Intelligent driver model

In traffic flow modeling, the intelligent driver model (IDM) is a time-continuouscar-following model for the simulation of freeway and urban traffic. It was developed by Treiber, Hennecke and Helbing in 2000 to improve upon results provided with other "intelligent" driver models such as Gipps' model, which lose realistic properties in the deterministic limit.

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As a car-following model, the IDM describes the dynamics of the positions and velocities of single vehicles. For vehicle α{\displaystyle \alpha }, xα{\displaystyle x_{\alpha }} denotes its position at time t{\displaystyle t}, and vα{\displaystyle v_{\alpha }} its velocity. Furthermore, lα{\displaystyle l_{\alpha }} gives the length of the vehicle. To simplify notation, we define the net distancesα:=xα−1−xα−lα−1{\displaystyle s_{\alpha }:=x_{\alpha -1}-x_{\alpha }-l_{\alpha -1}}, where α−1{\displaystyle \alpha -1} refers to the vehicle directly in front of vehicle α{\displaystyle \alpha }, and the velocity difference, or approaching rate, Δvα:=vα−vα−1{\displaystyle \Delta v_{\alpha }:=v_{\alpha }-v_{\alpha -1}}. For a simplified version of the model, the dynamics of vehicle α{\displaystyle \alpha } are then described by the following two ordinary differential equations:

Free road behavior: On a free road, the distance to the leading vehicle sα{\displaystyle s_{\alpha }} is large and the vehicle's acceleration is dominated by the free road term, which is approximately equal to a{\displaystyle a} for low velocities and vanishes as vα{\displaystyle v_{\alpha }} approaches v0{\displaystyle v_{0}}. Therefore, a single vehicle on a free road will asymptotically approach its desired velocity v0{\displaystyle v_{0}}.

Behavior at high approaching rates: For large velocity differences, the interaction term is governed by −a(vαΔvα)2/(2absα)2=−(vαΔvα)2/(4bsα2){\displaystyle -a\,(v_{\alpha }\,\Delta v_{\alpha })^{2}\,/\,(2\,{\sqrt {a\,b}}\,s_{\alpha })^{2}=-(v_{\alpha }\,\Delta v_{\alpha })^{2}\,/\,(4\,b\,s_{\alpha }^{2})}.

This leads to a driving behavior that compensates velocity differences while trying not to brake much harder than the comfortable braking deceleration b{\displaystyle b}.

Behavior at small net distances: For negligible velocity differences and small net distances, the interaction term is approximately equal to −a(s0+vαT)2/sα2{\displaystyle -a\,(s_{0}+v_{\alpha }\,T)^{2}\,/\,s_{\alpha }^{2}}, which resembles a simple repulsive force such that small net distances are quickly enlarged towards an equilibrium net distance.

This comparison shows that the IDM does not show extremely irrealistic properties such as negative velocities or vehicles sharing the same space even for from a low order method such as with the Euler's method (RK1). However, traffic wave propagation is not as accurately represented as in the higher order methods, RK3 and RK 5. These last two methods show no significant differences, which lead to conclude that a solution for IDM reaches acceptable results from RK3 upwards and no additional computational requirements would be needed. Nonetheless, when introducing heterogeneous vehicles and both jam distance parameters, this observation could not suffice.